I tried as you suggested but didn't notice a significant difference. I will keep it in mind for the future though, so I appreciate the suggestion!
However, I did notice that when I was powering the setup from the computer alone via USB, it performed as expected. When I powered the setup from both the power supply and the computer, the data was noisy. I've given examples of a touch in both situations. As you can see, the touch when powered by USB is clean and clear as expected, while with the power supply it goes up and down.
Powered by USB
Powered by the power supply
So, I tried another power supply and got expected results! However, it was 12 V so I had to run it through a buck converter. I wondered if the buck converter had a "cleaning" effect on the electricity, so I powered the project using the original power supply unit but running it through the buck converter and again got bad results.
Am I correct in concluding that among these cheap switching power supply units, some just produce "dirty" power and aren't suitable for powering capacitive touch projects? If so, is there a way to know in advance whether they will be suitable or not? They aren't terribly expensive, but if I have to buy 10 to find one good one, it's not very economical.
Thank you as always for your kind response!